It is pretty clear that one cannot use a second board to "mirror" the current position in an OTB tournament. Anyone who disagrees is invited to try this at any official USCF tournament and observe how fast the TD throws you out for cheating.
It is less clear whether this is allowed under chess.com's rules. Folks who argue that this is allowable analogize to use of a Monroi device in OTB chess, with the physical chess board being the equivalent to a Monroi. However, the use of a Monroi is an exception to the general rule against using any aids to help visualize the board position, and is only allowed because the Monroi enables notation. Use of a physical board in an online match is not necessary to enable notation (as this is done automatically by the computer) and serves not purpose than to help the player using the board to visualize (and, yes, analyze) the current game.
If use of a physical board is allowed, then the same restrictions that applies to Monroi devices must also be imposed -- namely, the move must be made on the computer (which is the official board) before the move is made on the physical board. Moving the piece first on the physical board would therefore result in disqualification.
...but when I get access to my rulebook I'll look and see if there is anything in there about it in the section for Internet play.
In Chapter 10, of the 6th edition USCF rulebook, where it talks about general Internet match/tournament play there is consideration for physical boards being used. In such an instance is is suggested an assitant TD tranfer the moves from the interface to the board but it is not forbidden to use a physical board.